A350871 Number of well-rounded sublattices of index n in square lattice.

1, 1, 0, 1, 2, 0, 0, 1, 1, 2, 0, 2, 2, 0, 2, 1, 2, 1, 0, 2, 0, 0, 0, 4, 3, 2, 0, 0, 2, 2, 0, 1, 0, 2, 2, 1, 2, 0, 0, 4, 2, 0, 0, 0, 2, 0, 0, 4, 1, 3, 0, 2, 2, 0, 0, 0, 0, 2, 0, 8, 2, 0, 2, 1, 4, 0, 0, 2, 0, 2, 0, 1, 2, 2, 4, 0, 2, 0, 0, 6, 1, 2, 0, 2, 4, 0, 0

Offset: 1

Author

Andrey Zabolotskiy, Jan 20 2022

Keywords

• nonn

Comments

A sublattice is well-rounded if the linear span of its vectors of minimal length is the whole space.

A sublattice of the square lattice is well-rounded when it is square or centered rectangular (rhombic) with not too oblong unit cell: the angles of the rhombus should be at least Pi/3.

In this sequence, any two sublattices differing by any isometry are counted as distinct.

Examples

			a(4) = 1 well-rounded index-4 sublattice has basis (2, 0), (0, 2).
a(5) = 2 w.-r. index-5 sublattices have bases (2, 1), (-1, 2) and (1, 2), (-2, 1).
At index 12, for the first time centered rectangular sublattices occur, there are a(12) = 2 of them with the bases (3, 2), (3, -2) and (2, 3), (-2, 3).
		

Links

• Andrey Zabolotskiy, Table of n, a(n) for n = 1..10000
• Michael Baake and Peter Zeiner, Geometric enumeration problems for lattices and embedded Z-modules, arXiv:1709.07317 [math.MG], 2017; in: Aperiodic Order, vol. 2: Crystallography and Almost Periodicity, eds. M. Baake and U. Grimm, Cambridge University Press, Cambridge (2017), pp. 73-172. See table "Some counts of the enumeration problems for Z^2"; beware of the typo in the 60th term.
• Peter Zeiner, Coincidence Site Lattices and Coincidence Site Modules, Thesis, Universität Bielefeld, 2015.
• Index entries for sequences related to sublattices
• Index entries for sequences related to square lattice

Crossrefs

Cf. enumeration of other classes of sublattices of Z^2: A000203 (all sublattices), A002654 (square sublattices), A000089 (primitive square sublattices), A350872 (coincidence sublattices), A145393 (all sublattices up to isometries of the parent lattice).

Cf. A097584.

Programs

Mathematica

fa[s_] := Count[Divisors[s], _?(#^2 < (s/#)^2 < 3 #^2 &)];
f0[s_] := If[OddQ[s], 0, 2 fa[s/2]];
f1[s_] := With[{e2 = IntegerExponent[s, 2]}, 2 (-1)^e2 fa[s/2^e2]];
pr[s_] := Count[Range[s], _?(Mod[#^2 + 1, s] == 0 &)]; (*A000089*)
sq[n_] := Sum[pr[n/d^2], {d, Select[Range[n], Mod[n, #^2] == 0 &]}]; (*A002654*)
a[n_] := sq[n] + Sum[pr[n/d] (f0[d] + f1[d]), {d, Divisors[n]}];
Array[a, 87]

Formula

See Zeiner (2015), Theorem 7.3.1. [Note that the formula from Baake & Zeiner (2017) contains an error.]